NEW DELHI—Unable to spend or deposit their sackfuls of large bank notes amid India’s crackdown on hoarding cash, business owners across the country are paying employees months of salary in advance, ringing up bogus sales and even buying gold they can smuggle overseas to get rid of stashed money or conceal its source.

Such illegal workarounds are threatening to undercut Prime Minister Narendra Modi’s move this month to cancel India’s highest-denomination rupee bills, which was meant to punish tax evaders and other criminals and bring more of the nation’s $2 trillion economy out of the shadows.

If Mr. Modi’s unprecedented social-engineering project fails to net too many of the biggest tax cheats, he risks further incurring the wrath of Indians already frustrated with the pain and economic dislocation the experiment has brought about in its first two weeks.

“Canceling these 500- and 1,000-rupee notes has caused inconvenience to you,” the prime minister said at a recent rally. “But some people’s whole life has been ruined—that is how I have punished them. Because they looted the poor, the middle class. They looted your money to run their business. That is why I launched this fight.”

Tax officials in India have for decades played testy games of cat-and-mouse with rich individuals and businessmen who accumulate wealth off the books and store it as real estate, jewelry, financial assets and cash stuffed in wardrobes.